Hopf bifurcation of hybrid van der Pol oscillators
The authors study the dynamics of the modified Van der Pol equation \[ \ddot{x} + \varepsilon \biggl(\biggl(x^2 + {\dot{x}^2 \over \mu^2}\biggr)-\rho^2\biggr) \dot{x} + \mu^2 x = 0 \] subject to the constraint \(x \geq 0\). When \(x(t)\) hits the boundary, an impact with restitution coefficient \(e\) occurs: \[ \dot{x}(t^+) = -e \dot{x}(t^-). \] The authors prove that for certain values of the parameter \(\alpha=\varepsilon\rho^2/\mu\) the origin loses stability and a periodic solution bifurcates at this Neimark-Sacker bifurcation. Using some numerical software, the authors calculate the Neimark-Sacker bifurcation boundary in the \((e, \alpha)\)-plane and provide a (quite inaccurate) interpolation formula. It should be noted, that this stability boundary could be given explicitely by \[ \alpha_c = {-2\log e \over \sqrt{\pi^2 + \log^2 e}}. \] Finally, the authors demonstrate that by varying the parameters different solution amplitudes can be enforced.
scientific article; zbMATH DE number 6807803